3.1.1. Fuzzy Set

A fuzzy set is used to address the issues that exist in an uncertain environment. It was proposed by Zadeh when attempting to solve multiple criteria decision-making problems. If the set *Z* & is a triangular fuzzy number (TFN), each value of the membership function is in the range [0, 1]:

$$\mu\_{\widehat{Z}}(\mathbf{x}) = \begin{cases} \frac{(\mathbf{x} - \mathbf{j})}{(k - j)} & j \le \mathbf{x} \le k \\\ \frac{(l - \mathbf{x})}{(l - k)} & k \le \mathbf{x} \le l \\\ 0 & \end{cases} \tag{1}$$

Each value of membership, which includes the left (the lowest value to evaluate) and right (the highest value to evaluate) representatives, of a TFN is shown below:

$$\overline{\mathbf{N}} = (\mathbf{N}^{1(\mathbf{y})}, \mathbf{N}^{r(\mathbf{y})}) = (\mathbf{j} - (\mathbf{k} - j)\text{ y, } l + (\mathbf{k} - d)\text{ y}), \text{ y} \in [0, 1].$$

A triangular fuzzy number can be described as in Figure 4.

**Figure 4.** A triangular fuzzy number (TFN).
